Simplify the expression $(3k^4)^4$
Understand the Problem
The question asks us to simplify the expression. To do this, we will do the following: first, distribute the exponent and then simplify.
Answer
$81k^{16}$
Answer for screen readers
$81k^{16}$
Steps to Solve
- Distribute the exponent
Apply the power of a product rule: $(ab)^n = a^n b^n$ $$(3k^4)^4 = 3^4 \cdot (k^4)^4$$
- Simplify $3^4$
Calculate $3^4$: $$3^4 = 3 \cdot 3 \cdot 3 \cdot 3 = 81$$
- Simplify $(k^4)^4$
Apply the power of a power rule: $(a^m)^n = a^{m \cdot n}$ $$(k^4)^4 = k^{4 \cdot 4} = k^{16}$$
- Combine the terms
Combine the simplified terms: $$3^4 \cdot (k^4)^4 = 81 \cdot k^{16} = 81k^{16}$$
$81k^{16}$
More Information
The power of a product rule and the power of a power rule are crucial for simplifying expressions with exponents.
Tips
A common mistake is forgetting to apply the outer exponent to the coefficient (3 in this case). Remember to apply the exponent to all parts of the term inside the parentheses. Also, a frequent error is adding the exponents when you should multiply them.
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